Integration

MATH1051

9.1 Antiderivatives

A function $F$ is called an antiderivative of $f$ on an interval $I$ if

$$F^{\prime }(x)=f(x),\text{ for all }x\in I.$$

9.5 Volume of Revolution

Suppose $f(x)$ is positive and continuous on $[a,b]$. By rotating the graph of $f(x)$ above $[a,b]$ about the $x$-axis, we obtain a cylindrical solid. What is the volume of such a solid? This is what we call a volume of revolution.

9.5.1 Formula for the Volume of Revolution

Suppose $f(x)\ge 0$ and continuous on $[a,b]$. The volume of revolution of the solid obtained by rotating the graph $y=f(x)$ above $[a,b]$ about the $x$-axis is:

$$\mathbf{V}=\pi {\int }_a^b[f(x){]}^{2}dx$$

9.5.3 Example

Let $a>0$. Find the volume of the solid obtained by rotating $y=f(x)=\sqrt{x}$ about the $x$-axis over $[0,a]$.

$$V=\pi {\int }_0^a{\left[\sqrt{x}\right]}^{2}dx$$ $$\begin{array}{rl}& =\pi {\int }_0^{a}xdx=\pi {\left[\frac{x^2}{2}\right]}_0^a\\ & =\frac{1}{2}\pi a^2\end{array}$$